Differentiation

A Derivative of a function is the instantaneous rate of change of that function at a point. It's the limit as the change in x approaches zero of the average slope of a graph. The operation of taking a derivative of a function can be written in different notation, like f"(x) or d(f(x))/dx . The rules for taking derivatives of some basic functions are outlined below:

if n is a constant and u and v v are functions of x
                                                 
when f(x)=                                             |                                                    f '(x)=
un                                                           |                                             u'(x)ᐧ (n)un-1 
nu                                                           |                                             u'(x)ᐧln(n)ᐧnu
n                                                             |                                                 0
uᐧv                                                          |                                          (u'(x)ᐧv)+(v'(x)·u)
u                                                             |                                        (u'(x)ᐧv)-(v'(x)·u)
v                                                             |                                                   v2 
u(v(x))                                                    |                                          u'(v(x))ᐧv'(x) (d(u(v(x)))/(v(x)) )(v'(x))
sin(x)                                                      |                                                   cos(x)
cos(x)                                                      |                                                   -sin(x)
tan(x)                                                     |                                                   (sec(x))2
cot(x)                                                     |                                                    -(csc(x))2
arcsin(x)                                                 |                                                 u'(x)ᐧ    1    
                                                               |                                                        √1-u2
arccos(x)                                                 |                                                     -u'(x)ᐧ    1    
                                                               |                                                               √1-u2
arctan(x)                                                 |                                                     u'(x)ᐧ    1    

                                                               |                                                               1+u2
ln(x)                                                       |                                                       1/x
log base a (x)                                         |                                                     1/x(ln(a))

to differentiate a function like xx,you need to use the natural log rule: ln(xu) =uᐧln(x).
When given a table of values you can estimate the derivative at a point at or in between input values given.

If a function is differentiable on an interval then the function is continuous on the interval but if the  function is continuous on the interval then the function is not necessarily differntiable ( i.e. functions can be non-differentiable and continuous ( with sharp cusps and the like ).

The MVT states that if a function is continuous and differentiable on an interval [a,b] then the is some point c where f '(c) is equal to the slope between a and b or the average rate of change over the interval. Rolles theorem is a special case of the mean value theorem when the average rate of change is 0.


Do differentiate parametric functions you take the derivative of y with respect to t and then divide id by the derivative of x with respect to t.

To differentiate polar functions you use the definitions y=rsinθ and x=rcosθ and then use parametric differentiation.

Two take second derivatives ( f "(x) ) you take the derivative of the derivative, to take third derivatives you take the derivative of the derivative of the derivative or the derivative of the second derivative and so on.

To take derivatives of inverse functions at a point, you take the reciprocal of the derivative of the corresponding point.

When a derivative of a function is undefined or equals zero you have got a critical point of that function. You can use derivatives to determine if a function is increasing( i.e. the derivative is positive ), or decreasing ( negative f '(x) ), concave up or concave down ( the second derivative is positive for concave up and negative for concave down. ). These determinations can be used to find maxima or minima of functions. They can also be used to find where the concavity of the graph changes ( a inflection point). There are multiple methods to do this including finding critical points and evaluating the second derivative at the points ( second derivative test, if it is positive it is a local minimum , if it is negative it is a local maximum ), or evaluating f ' before and after the critical point.
Finding local maxima and minima and points of inflection are all part of derivative applications, like useful thing you can use differential Calculus for. Approximating the value of a function using slope tangent line approximation is also part of Derivative applications. Basically you use the equation
y-y1= f '(x1)ᐧ(x-x1). Where you can approximate y values ( the y ) of x values ( the x ) near the  x1 value. The y1  is the y value at the x1.

Another application of derivatives is related rates. Which is when you have a object, and some measurement of it is changing at a given constant rate, while some other property or properties is/are changing at a non-constant rate. You use an equation that relates the two properties and then solve it for the rate of change of the non-constant one at some point in time.

One application of differential calculus has a bunch to do with Physics, namely the analyzation and study of the movement of a particle given its function of either , position, velocity, or acceleration with time. The derivative of position is velocity, the derivative of velocity is acceleration. Which means that the slope of a position versus time graph is the velocity and the slope of a velocity versus time graph is the acceleration. These definitions can help us analyze and determine when a particle is moving in a certain direction , when it is not moving, when it is slowing down or speeding up ( when the acceleration and velocity are in the same direction then the particle is speeding and when the velocity and acceleration are in different directions the particle  is slowing ), and how fast or how much a particle is accelerating at a point.

Differential calculus can also be used to solve optimization problems, in which you write a function for a thing to me maximized or minimized and then take the derivative(s) of that function to find the local an global maxima or minima.